Atticus taught her that you never really know a person until you "climb into their skin" and walk around in it. In other words, you've got to climb into their shoes and see things from their perspective. By teaching her how to understand the views of others, Atticus is preparing her to understand life in a unique and mature way.

This is particularly important considering the drama that will begin to unfold in the town as Tom Robinson's trial takes place. People who Scout thought were good will go "stark-raving mad" when the trial begins, because it involves race at the core of the subject matter.

Sterotypically, the South has always had race problems. While it is true that a lot of documented racial persecution and strife has taken place in the Southern United States, it is also true that all areas experience some sort of racial tubulance from time to time. This is important to note as one reads the story, so as to avoid forming a negative opinion of the South as a whole. Atticus even tries to explain that the people are genuinely good, but when race becomes an issue, their perspective changes nearly 100%.

Scout learns that in order to fully understand people, she must put herself in their position. This is what Atticus means by his statement about "walking around" in another person's skin.
An old American Indian proverb states that one may not judge another until they've walked a mile in that other person's "moccasins," and the line of thinking that Atticus is giving Scout reiterates that idea.
Scout sees people she knows well exhibiting prejudice and bigotry, and she doesn't know how to handle it until Atticus "clears the air" for her, using this piece of fatherly advice.